Single ended circuits operate in class A, that includes all regular pre-amp / phase splitter circuits; due to non linearities in the transfer function, these tend to generate even harmonics (ie asymmetric output waveforms), this tendency increasing with signal level.
Push-pull circuits tend to cancel out even harmonics that they create, whether they operate in class A, AB or B.
If you are coming to this from a 'even harmonics = good distortion, odd harmonics = bad distortion' viewpoint, my thinking (with respect) is that it's nonsense; best to ignore it.
Rather, my take is that low order harmonics tend to sound smoother than high order harmonics.
Pleasant sounding distortion tends to be due to low order harmonics being generated before higher order harmonics (as the signal level rises).
You may find the http://www.valvewizard.co.uk/gainstage.html download useful, as indeed that whole site.
Pete

Many vendors claim that if you buy the part they are hawking you'll get "blues/rock/sweet/rock/Hendrix/SRV/Clapton/God's/vintage/whatever" sound but truth is , parts interact with each other.

In the case of overdriven tubes, biasing and available plate voltage have *a lot* to say in the final waveshape, mainly because you will be working in different places of a very non-linear and assymetric graph.

For pre-amps, I've no experience other than 12AX7 in stages which are likely to get overdriven, but as there's a wide variety of overdrive characters available from that, my guess is that the particular circuit, design topology and implementation are probably very significant.
I intend to tinker with EF86 sometime.
For power amps, it's difficult to separate the tube type from the circuit they tend to be used in; generally, higher plate voltage leads to a brighter, perhaps harsher overdriven tone. Also, since the end of mainstream tube production, there's seems very little control / regulation over tube conformance to type, so eg a 6L6GC can be anything that will sort-of work in normal such applications.
Pete

Do some tubes create more lower order harmonics than others when driven into clipping or should I say less high order harmonics? Or is it more surrounding circuit design, and/or other factors?

+1 to what everybody else said about circuit design. Let me add that in my experience, a *small* amount of distortion of the signal (compression, a little clipping) will result in a small amount of overtones being generated, and they will be predominantly lower-order harmonics. Add another stage, and another, that all produce a little distortion, and the lower-order harmonics start to add up.
A single stage, MASSIVELY overdriven, will clip much harder than the example above, and produce many overtones, including the higher-order harmonics. When multiple stages are considered I'm guessing that intermodulation distortion can multiply the effect, but that's speculation. Most amps that "splatter" the tone like this do it for effect - the 'whole package' mentioned above - and will filter out some of the low frequency information (at some point in the signal path) so that the overall effect is of a brighter, buzzier tone.

If you are coming to this from a 'even harmonics = good distortion, odd harmonics = bad distortion' viewpoint, my thinking (with respect) is that it's nonsense; best to ignore it.
Rather, my take is that low order harmonics tend to sound smoother than high order harmonics.
Pleasant sounding distortion tends to be due to low order harmonics being generated before higher order harmonics (as the signal level rises).

I've always heard the odd even argument but I have been using solid state guitar amps since the 60s and I'm tying to figure out why so many players prefer Tube amps. I'm sure style of music plays a part. Great link. Over my head at the moment. As with any new area of learning you have to learn the language as you go.

"The absence of even harmonics is more attributable to circuit topology (e.g., amount of feedback, amplifier gain, number of stages) than it is to circuit technology (e.g., tube or solid-state). However, because of the small size and low cost of solid-state components, it is common for them to be packaged into op-amps, where they are used with high levels of negative feedback. These systems tend to have odd-symmetric relationships between input and output signals. Because of this, absence of even harmonics has often been (wrongly) attributed to solid-state technology."

In order to visualize how and under what specific engineering conditions a tube will begin to distort, find a tube data sheet for the specific preamp tube that you are interested in, and look at the graph of the constant bias lines on an anode current vs anode voltage chart. A discussion of load lines can be found in many places on the internet that will walk you through how to read the chart.

A load line drawn from a HV supply of 300vdc (x-axis) diagonally across to 3ma current (y-axis) will be typical of a 12aX7 tube and is a good starting point for reading these graphs. Where that load line intersects the constant-bias lines is where the tube will 'sit' in a quiescent state. The distortion that you generate is dependent mostly on the amount of AC voltage signal that is fed into that tube stage. With a little 'over-drive', the grid begins to clip because it is feeding current (instead of voltage) as it tries to become positive and the signal 'cuts off' when too negative. On the chart, 'too negative' is the region at the bottom of the graph where the constant-bias lines all bunch up together.

With the grid (bias) voltage at 1/2 of the available 'swing', the AC signal starts to become audibly distorted (starts to clip) about when the AC peak voltage is equal to the bias voltage. At that point, mostly low-order harmonics are produced. If the AC signal is many times the grid bias, then the clipping is very heavy, and the output starts to take on the proverbial square wave shape.

A bias point that is not close to 1/2 the available range is called warm or cold, and will clip one side of the wave before the other. This can facilitate the production of even overtones, and is used for a variety of voicings and effects. But what Jazz P said, class A is where the tube is biased with the intent that all the signal going through the tube is all that you want to hear, that is the "whole wave".

'In tube clipping is there a top limit when the grid is too positive?'
Generally, the grid voltage isn't forced to be more positive that the cathode; when Vg-k > 0V, the grid characteristics change considerably and the grid effectively becomes a low impedance path to the cathode.
So the grid of say a 12AX7, from a static bias level of -2V, may accommodate the signal swinging up to 0V and down to -4V (voltages with respect to the cathode), in its linear area.

'Isn't the preamp where you want to generate distortion?'
For most classic guitar amps, the pre-amp circuits are intended to operate in their linear area.
Given realistic control settings, as signal level is increased, the power tubes will saturate (ie reach grid conduction) well before earlier stages have run out of 'headroom'.
Pete